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Is there a simple description of a Chow ring of a blow-up of a point on a smooth projective variety? Or at least of successive blow-ups of $\mathbb{P}^n$?
Maybe something like $A(\tilde{X})=f^*(A(X))\oplus\mathbb{Z}(E)$, where $f\colon\tilde{X}\to{}X$ is a blow-up, E is an exceptional divisor, with multiplication given by $E\cdot{}E_k=-E_{k-1}$, $E_0=f^*(P)$, where $E_k{}$ is a k-dimensional linear subspace of an exceptional divisor $E(=E_{n-1})$, and $P$ is a point we are blowing up. What I'm suggesting is true for surfaces (exercise 6.5 in appendix A of Hartshorne), and seems geometrically plausible in the case $X=\mathbb{P}^n$.
Also, it'd be great to know what cycles are effective.
I'm afraid all this is really trivial for someone understanding Fulton's book, but I'm not at that level yet.

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That's not exactly so, because n-th powers of $f^*(H)$ and $E$ are certainly identified. My question was more about how do I see that there are no other identifications. – 14555 Jan 14 '11 at 1:01
up vote 6 down vote accepted

The general formula about the intersection ring of blow-ups is discussed in Fulton's book. In your case you want to study the intersection ring of a smooth algebraic variety $V$ blown up at a point $Z$. There is a simple formula for this situation by Keel. You can find it in his paper: Intersection Theory of Moduli Space of Stable N-Pointed Curves of Genus Zero.

Another nice reference is the paper "A compactification of configuration spaces" by Fulton-MacPherson. In section 5 of this paper they mention the Keel's formula and state the facts needed in the computation of the Chow ring. I summarize it below.

The key fact is that the restriction map from the Chow ring of the variety $V$ to the Chow ring of the point $Z$ is surjective. The intersection ring of the blow-up $\widetilde{V}$ is generated over $A(V)$ by the class of the exceptional divisor $E$ with the ideal $I$ of relations described bellow:

1) Let $J_{Z/V}$ be the kernel of the restriction map from $A(V)$ to $A(Z)$. It contains all elements in $A(V)$ of positive degree, for example.

2) Assume that you can write $Z$ as a transversal intersection $\cap_{i=1}^r D_i$ of the divisor classes $D_i$. Define the polynomial $P_{Z/Y} \in A(V)[t]$ by the rule $P(t)=\prod_{i=1}^r(t+D_i)$. This polynomial is called a Chern polynomial of $Z$. It depends on the choice of the divisor classed $D_i$. It means that it is not unique and is determined up-to an element in $J_{Z/V}$.

The ideal $I$ is generated by $J_{Z/V}\cdot E$ and $P_{Z/V}(-E)$. The Chow ring of $\widetilde{V}$ is therefore equal to $\frac{A(V)[E]}{I}$.

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